SOME OPERATOR INEQUALITIES FOR UNITARY INVARIANT NORMS ∗†

Let L(H) be the algebra of bounded operators on a complex separable Hilbert space H. Let N be a unitary invariant norm defined on a norm ideal I ⊆ L(H). Given two positive invertible operators P,Q ∈ L(H) and k ∈ (−2, 2], we show that N(PTQ−1 + P−1TQ+ kT) ≥ (2 + k)N(T), T ∈ I. This extends Zhang’s inequality for matrices. We prove that this inequality is equivalent to two particular cases of itself, namely P = Q and Q = P−1. We also characterize those numbers k such that the map Υ: L(H) → L(H) given by Υ(T) = PTQ−1+P−1TQ+ kT is invertible, and we estimate the induced norm of Υ−1 acting on the norm ideal I. We compute sharp constants for the involved inequalities in several particular cases.